Mrs. Smith's Algebra 2 class scored very well on the last quiz. With one exception, everyone received an A. Within how many standard deviations of the mean do all the quiz grades fall? 97, 95, 97, 94, 93, 99, 91, 96, 94, 88, 92, 91 (1 point) Responses 1 1 4 4 3 3 2
To find the standard deviation, we first need to find the mean of the quiz grades.
(97 + 95 + 97 + 94 + 93 + 99 + 91 + 96 + 94 + 88 + 92 + 91) / 12 = 94.66666667 (approximately 94.67)
Next, we calculate the differences between each quiz grade and the mean:
2.33, 0.33, 2.33, -0.67, -1.67, 4.33, -3.67, 1.33, -0.67, -6.67, -2.67, -3.67
Then, we square these differences:
5.4289, 0.1089, 5.4289, 0.4489, 2.7889, 18.7489, 13.4689, 1.7689, 0.4489, 44.4889, 7.1289, 13.4689
Next, we calculate the average of these squared differences:
(5.4289 + 0.1089 + 5.4289 + 0.4489 + 2.7889 + 18.7489 + 13.4689 + 1.7689 + 0.4489 + 44.4889 + 7.1289 + 13.4689) / 12 ≈ 8.83833333
Finally, we take the square root of this average to find the standard deviation:
√8.83833333 ≈ 2.97515343
Since all quiz grades fell within 3 standard deviations of the mean (94.67 ± 3*(2.97515343) ≈ 94.67 ± 8.92666029), we can conclude that all the quiz grades fall within 3 standard deviations of the mean.